Tricky function

Algebra Level 3

The domain of the function $f(x) = \frac{x}{\sqrt{\sin(\ln { x } ) - \cos(\ln { x } )}}$ is

(e^{(2n+\frac{1}{4})\pi },e^{(3n-\frac{3}{4})\pi })

(e^{(2n+\frac{1}{4})\pi },e^{(2n+\frac{5}{4})\pi })

(e^{(2n\pi },e^{(3n+\frac{1}{2})\pi })

(e^{(2n+\frac{1}{2})\pi },e^{(2n+\frac{9}{4})\pi })

1 solution

Akhilesh Prasad
Sep 28, 2014

For the domain to be defined, the following conditions must satisfy

$\sin(\ln{x})>\cos(\ln{x})$ and $x>0$ ( $\because$ The argument of a logarithm can not be negative)

Hence, $2n\pi + \frac{\pi}{4}< \ln{x}< 2n\pi + \frac{5\pi}{4}$

$\Rightarrow$ $e^{(2n\pi + \frac{\pi}{4})} < x < e^{(2n\pi + \frac{5\pi}{4})}$

$\therefore$ $x \in (e^{(2n\pi + \frac{\pi}{4})},e^{(2n\pi + \frac{5\pi}{4})})$